Certain Number-Theoretic Episodes in Algebra

R. Sivaramakrishnan

Publisher:

Chapman & Hall/CRC

Publication Date:

2007

Number of Pages:

632

Format:

Hardcover

Series:

Pure and Applied Mathematics 286

Price:

139.95

ISBN:

0-8247-5895-1

Category:

Textbook

BLL Rating:

BLL*

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

MAA Review
Table of Contents

[Reviewed by

Mark Bollman

, on

05/18/2007

]

I have taught at three colleges, and have never been at a school with a regularly-offered number theory course. As a number theorist asked to teach undergraduate abstract algebra for the first time several years ago, I was drawn to a particular textbook (Hillman & Alexanderson) by its final chapter, which was explicitly about number theory. Sivaramakrishnan’s work would be an excellent followup to that kind of algebra course.

Beginning with the premise that “it is desirable to learn algebra via number theory and to learn number theory via algebra,” this book gives a thorough treatment of both subjects and clearly shows how each illuminates the other. If a number-theoretic result has an accessible algebraic analog, it is clearly presented in its generality. At the same time, the book includes fine coverage of more elementary number theory from this advanced vantage point — the Fibonacci sequence, Fermat’s Little Theorem, and Goldbach’s conjecture are all here among the discussions of topologies, Dedekind domains, and lattices.

While not for beginners in either subject — by design, of course — this book does an excellent job of cataloging and explaining the many dimensions of the rich relationship between algebra and number theory. To read it is a great challenge, as one might expect from such a far-reaching work, but one which amply rewards careful effort.

Mark Bollman ([email protected]) is an assistant professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.

ELEMENTS OF NUMBER THEORY AND ALGEBRA
Theorems of Euler, Fermat and Lagrange
Historical perspective
Introduction
The quotient ring Z / rZ
An elementary counting principle
Fermat's two squares theorem
Lagrange's four squares theorem
Diophantine equations
Notes with illustrative examples
Worked-out examples

The Integral Domain of Rational Integers
Historical perspective
Introduction
Ordered integral domains
Ideals in a commutative ring
Irreducibles and primes
GCD domains
Notes with illustrative examples
Worked-out examples

Euclidean Domains
Historical perspective
Introduction
Z as a Euclidean domain
Quadratic number fields
Almost Euclidean domains
Notes with illustrative examples
Worked-out examples

Rings of Polynomials and Formal Power Series
Historical perspective
Introduction
Polynomial rings
Elementary arithmetic functions
Polynomials in several indeterminates
Ring of formal power series
Finite fields and irreducible polynomials
More about irreducible polynomials
Notes with illustrative examples
Worked-out examples

The Chinese Remainder Theorem and the Evaluation of Number of Solutions of a Linear Congruence with Side Conditions
Historical perspective
Introduction
The Chinese Remainder theorem
Direct products and direct sums
Even functions (mod r)
Linear congruences with side conditions
The Rademacher formula
Notes with illustrative examples
Worked-out examples

Reciprocity Laws
Historical perspective
Introduction
Preliminaries
Gauss lemma
Finite fields and quadratic reciprocity law
Cubic residues (mod p)
Group characters and the cubic reciprocity law
Notes with illustrative examples
A comment by W. C. Waterhouse
Worked-out examples

Finite Groups
Historical perspective
Introduction
Conjugate classes of elements in a group
Counting certain special representations of a group element
Number of cyclic subgroups of a finite group
A criterion for the uniqueness of a cyclic group of order r
Notes with illustrative examples
A worked-out example
An example from quadratic residues

THE RELEVANCE OF ALGEBRAIC STRUCTURES TO
NUMBER THEORY
Ordered Fields, Fields with Valuation and Other Algebraic Structures
Historical perspective
Introduction
Ordered fields
Valuation rings
Fields with valuation
Normed division domains
Modular lattices and Jordan-Hölder theorem
Non-commutative rings
Boolean algebras
Notes with illustrative examples
Worked-out examples

The Role of the Möbius Function-Abstract Möbius Inversion
Historical perspective
Introduction
Abstract Möbius inversion
Incidence algebra of n × n matrices
Vector spaces over a finite field
Notes with illustrative examples
Worked-out examples

The Role of Generating Functions
Historical perspective
Introduction
Euler's theorems on partitions of an integer
Elliptic functions
Stirling numbers and Bernoulli numbers
Binomial posets and generating functions
Dirichlet series
Notes with illustrative examples
Worked-out examples
Catalan numbers

Semigroups and Certain Convolution Algebras
Historical perspective
Introduction
Semigroups
Semicharacters
Finite dimensional convolution algebras
Abstract arithmetical functions
Convolutions in general
A functional-theoretic algebra
Notes with illustrative examples
Worked-out examples

A GLIMPSE OF ALGEBRAIC NUMBER THEORY
Noetherian and Dedekind Domains
Historical perspective
Introduction
Noetherian rings
More about ideals
Jacobson radical
The Lasker-Noether decomposition theorem
Dedekind domains
The Chinese remainder theorem revisited
Integral domains having finite norm property
Notes with illustrative examples
Worked-out examples

Algebraic Number Fields
Historical perspective
Introduction
The ideal class group
Cyclotomic fields
Half-factorial domains
The Pell equation
The Cakravala method
Dirichlet's unit theorem
Notes with illustrative examples
Formally real fields
Worked-out examples

SOME MORE INTERCONNECTIONS
Rings of Arithmetic Functions
Historical perspective
Introduction
Cauchy composition (mod r)
The algebra of even functions (mod r)
Carlitz conjecture
More about zero divisors
Certain norm-preserving transformations
Notes with illustrative examples
Worked-out examples

Analogues of the Goldbach Problem
Historical perspective
Introduction
The Riemann hypothesis
A finite analogue of the Goldbach problem
The Goldbach problem in Mn(Z)
An analogue of Goldbach theorem via polynomials over finite fields
Notes with illustrative examples
A variant of Goldbach conjecture

An Epilogue: More Interconnections
Introduction
On commutative rings
Commutative rings without maximal ideals
Infinitude of primes in a PID
On the group of units of a commutative ring
Quadratic reciprocity in a finite group
Worked-out examples

True/False Statements: Answer Key
Index of Some Selected Structure Theorems/Results
Index of Symbols and Notations
Bibliography
Subject Index
Index of names